Environmental Engineering Reference
In-Depth Information
when warmer fluid is introduced below a cooler layer. Friction factors can be calcu-
lated using correlations appropriate to both the nonisothermal thermally-destabilized
low Reynolds number flow and the isothermal developing laminar flow present in the
lengths of straight pipeline. Empirically-determined laminar heat loss coefficients for
the pipe bends are employed usually. Time variations of insolation, ambient temper-
atures and hot water withdrawal are the inputs to most simulation, models with the
transmission of the glass collector cover being a function of the sun-hour angle.
A two-dimensional finite difference approach takes account of the thermal capaci-
tance of the collector. In the derivation of the energy equations for the collector model,
a glass cover, opaque to long wave radiation, and parallel fin-and-tube collector plate
are treated usually as two large, parallel, grey bodies for the analysis of radiative heat
exchanges. In addition, the glass cover is assumed to be at a uniform temperature at
each moment in time and is therefore represented by a single node. It is also assumed
usuallythat the temperature gradient through each thin fin is constant so that two-
dimensional planar conduction is assumed to prevail. Conduction within the collector
fluid in the direction of the main flow is taken to be negligible in most analysis as
are the thermal capacities of the thermal insulations applied to the hot-water store,
collector, and connecting pipes. An energy balance on an incremental volume of the
fin gives, for a two-dimensional plate temperature distribution,
∂
2
T
f
∂x
2
∂
2
T
f
∂y
2
ρ
f
C
f
δ
f
∂T
f
∂t
=
k
f
δ
f
+
+
h
f .g
(
T
g
−
T
f
)
+
U
f
,
a
(
T
a
−
T
f
)
(
T
g
−
σ
T
f
)
+
+
(
T
α
)
e
I.
(4.4.4)
ε
−
1
+
ε
−
1
f
−
1
g
The boundary conditions are: (
i
) from symmetry of adjacent nodes
(
w
,
L
,
t
)
=
∂T
f
∂x
∂T
f
∂x
(
2
,
o
,
t
)
=
0
(4.4.5)
and,
(ii)
as there is no heat flux through ends of the plate
(
x
,
o
,
t
)
=
(
x
,
L
,
t
)
=
∂T
f
∂y
∂T
f
∂y
0
(4.4.6)
The boundary condition relating the temperatures of the fluid in the risers and
the plate is obtained from a heat balance or an incremental volume of the fluid. The
temperature of the pipe wall is assumed to be that of the fin at
x
=
0. Thus:
ρ
w
C
w
πD
r
∂T
w
∂t
C
w
m
c
N
∂T
w
∂y
+
=
h
rw
πD
r
×
(
T
f
(
o
,
y
,
t
)
−
T
w
)
.
(4.4.7)
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